ROUGH DRAFT authorea.com/62406
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  • Chirp manipulation

    Linear Chirp

    Time domain representation

    Zero centered baseband chirp:

    \[s(t) = e^{j\pi\gamma t^2}\] where \(t\) - time; \(\gamma = \frac{B}{\tau}\) - frequency slope; \(B\) - chirp bandwidth; \(\tau\) - chirp duration; \(T\) - sampling period; \(2M\) - number of samples.

    M = 100;
    n = -M : M-1;
    T = 0.1;
    t = n*T;
    gamma = 0.15;
    sChirp = exp(1j*pi*gamma*t.^2);
    plot_complex(t, sChirp, 'time')
    

    Complex chirp in time domain

    Frequency domain representation

    \[S(f) = \mathcal{F}(s(t)) = \int_{-\infty}^{\infty } {e^{j\pi\gamma t^2} e^{-j2\pi f t}dt}\]

    SChirp = fft(sChirp);
    f = (0:2*M-1)/(T*2*M);
    plot_complex(f, SChirp, 'freq')